Using a square, or something similar,
mark a square around the hole and cut it out.

Cut drywall from behind

Step 2:

This makes it easier to cut another piece of drywall to patch the hole. When
cutting the hole, make sure you minimize damage to the back of the drywall (it
looks like this from behind). Cut
through the drywall and paper on the back.

Scrap wood

Step 3:

Cut a small scrap of wood (1x4 works good) slightly longer than the
hole. Being careful not to drop the board, insert it into the hole.

Screw block from behind

Step 4:

Use drywall screws to secure a the board (or similar material) behind
the drywall in the middle of the hole. Drive the screws so that they
are just slightly below the surface of the drywall. If the hole is
large you may need more than one brace.